The NFL Network first reported the Chiefs' pending deal with Foles.
The Rams granted quarterback Nick Foles' request to be released, announcing the move Wednesday.
The Chiefs had been among at least three teams that were a possible destination for Foles.
The Cowboys had discussions with Foles' agent about making Foles a possible replacement for the injured Kellen Moore. The Minnesota Vikings also expressed interested in Foles, a source confirmed to Graziano.
Foles will reunite with Chiefs coach Andy Reid, who was head coach of the Eagles when Philadelphia made Foles a third-round draft pick in 2012.
For his career, Foles has a 19-16 record with 53 touchdown passes and 27 interceptions. The Rams released him before training camp after being unable to swing a trade.
Information from ESPN's Todd Archer was used in this report.